The fourier series of periodic and real analytic function

Question

Let $f$ be a real analytic and periodic function defined on the interval $[0, 2\pi]$. Then $f$ is infinitely differentiable for sure. Therefore, the fourier coefficients of $f$ decay faster than any polynomial function of the integer.
So it is clear that the fourier series of $f$ converges aboslutely and uniformly. (Moreover, the series converges to $f$ pointwise.)

Am I correct?

Answer 1

Let $f$ be periodic. $\sum |\hat {f} (n)| <\infty$ implies that the Fourier series converges absolutely and uniformly. Since this condition is satisfied here (by comparison with $\sum\frac 1 {n^{2}}$) the answer is a definite YES.